$k$-arrangements, statistics and patterns

TL;DR

This paper studies $k$-arrangements, a permutation class with fixed points colored by $k$, providing enumerative results, confirming conjectures, and analyzing Eulerian statistics using the Decrease Value Theorem.

Contribution

It proves enumerative results for $k$-arrangements, confirms several conjectures on their statistics and patterns, and applies the Decrease Value Theorem to derive generating functions.

Findings

Confirmed conjectures on the distribution of descents in $k$-arrangements.

Derived generating functions for Eulerian statistics on permutations.

Strengthened previous results on the equdistribution of specific permutation statistics.

Abstract

The $k$-arrangements are permutations whose fixed points are $k$-colored. We prove enumerative results related to statistics and patterns on $k$-arrangements, confirming several conjectures by Blitvi\'c and Steingr\'imsson. In particular, one of their conjectures regarding the equdistribution of the number of descents over the derangement form and the permutation form of $k$-arrangements is strengthened in two interesting ways. Moreover, as one application of the so-called Decrease Value Theorem, we calculate the generating function for a symmetric pair of Eulerian statistics over permutations arising in our study.